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In a course of introduction to Generative Grammar, my teacher told that auxiliary (Aux in the tree diagram) is the element that comes before the subject (NP) in an interrogative sentence.

But I found this rule causes a trouble in the case of the sentence "John can be happy.", because only 'can' can be the (modal) auxiliary in the rule but I learned 'be' is also known as an auxiliary.

What is a more comprehensive rule to identify Aux in a given sentence in Generative Grammar?

And for that example, the diagram below represents the structure of the sentence, but how can it be written right?

enter image description here

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    Aux is a phrase, not just a category of words, though the term is also used as a word category. In your example sentence diagram, I would say "be" is part of the Aux "can be". It is only the first Aux constituent which appears before the subject in an interrogative. – Greg Lee Aug 1 '17 at 14:36
  • Preliminary point: "be" is virtually always an auxiliary verb. "Can be happy" is the VP, in which "can" is a catenative verb and the subordinate clause "be happy" is catenative complement of "can". – BillJ Aug 1 '17 at 18:16
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Your teacher seems to have referred to a very elementary and partial account of auxiliaries popular in the 1960's that does happen to handle the example you cite, since be is not an auxiliary in that case but a 'main' verb, but is entirely inadequate to describe the complexity of English in this area and has long been superseded by much more detailed and precise theories.

Nevertheless, you surely are aware that an English sentence may contain up to four auxiliary verbs, as in You could have been being followed by the police, where the first four 'verb-like' elements do belong to the class of auxiliaries, and elsewhere would satisfy all the relevant tests of auxiliarihood, but only the first one, the modal auxiliary could, which is finite/tensed, can be 'inverted' with the subject in the corresponding question or appear preceding not in the corresponding negative sentence.

In such internally more complex sequences, the 'verbs' after could (aspectual have, aspectual be, and 'passive' be) are also 'auxiliaries', and could themselves be 'raised' to Tense/Infl and be 'inverted' across the subject in questions and precede negation if they appeared first and carried tense, but, as things are, they can do neither, because in an English simple clause only one 'verb' can be tensed, whichever is leftmost (= hierarchically higher). The following ones, if any, will carry non-finite inflections selected by whichever other auxiliary precedes them (i.e., 'zero' after modals, past participle -ed/-en after aspectual have, present participle -ing after aspectual be, and passive past participle after 'passive' be), and, as a consequence, will not qualify to appear under Tense/Infl (the head of the predicative part of the clause) nor be able to undergo 'subject-auxiliary inversion' (further movement of the tensed auxiliary from Tense/Infl into a higher head), or to appear before negation.

Your question also reveals that you are, understandably, a bit confused by a terminological problem that arose in early Chomskian syntax: in early TGG (1950's and 60's), the label 'Aux' was used in two different ways: a) as a functional label for the node dominating all the auxiliary verbs a sentence contained, tensed or not (i.e., the 'auxiliary component' in opposition to the 'main verb phrase'), as in Chomsky's (1955) thesis, Syntactic Structures (1957), or when, even as late as in Aspects (1965), he continued to offer the PS rule Pred Phrase -> Aux + VP..., and b) in a more restrictive sense, to refer to whichever auxiliary carried tense and could invert with the subject, precede negation, and undergo a few other important 'transformations'.

In the latter sense of 'Aux', the tree diagramme you show us is almost 'right' (ignoring that Aux and VP do form a unique constituent and should be dominated by a single node), since that sentence contains just one auxiliary, can, and, of course, can is finite, but you should not assign the same NP + Aux + VP structure to sentences like my You could have been being followed by the police: there is massive syntactic and semantic evidence that a sentence like that cannot have the structure [[You] + [could have been being] + [followed by the police]]!

I suggest you consult some more recent English TG syntax textbook (maybe Haegeman & Guéron's English Grammar, or Radford's successive C.U.P. introductions, or Carnie's,... there are lots of them) to get a more up-to-date view of how the English auxiliary system is currently analysed, as well as of the syntactic, semantic, morphological, and phonological criteria by which English 'auxiliaries' can be distinguished from English 'main verbs'.

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    Better yet, consult McCawley 1998 to see what structures look like without all the sponsor labels. – jlawler Aug 1 '17 at 17:16

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